An Intrinsic Description of Shape

Abstract

Abstract.We give in this paper a description of the shape category of compacta in terms of multivalued maps. We introduce the notion of a multi-net and prove that the shape category of compacta is isomorphic to the category HN whose objects are metric compacta and whose morphisms are homotopy classes of multi-nets. This description is intrinsic in the sense that it does not make use of external elements such as ANR-expansions or embeddings in appropriate AR-spaces. We present many applications of this new formulation of shape. IntroductionShape theory was introduced in 1968 by K. Borsuk [3] as a new classification of compact metric spaces which takes into account their global properties and neglects the local ones. Borsuk's main notion was that of a fundamental sequence which plays the same role as the notion of mapping in homotopy theory. Fundamental sequences are defined in terms of the neighborhoods of compacta embedded in a convenient AR-space. Shape theory was later developed for more general spaces. Of special importance is the Mardesic-Segal approach [16] where shape theory is based on inverse systems of ANRs. In this approach, shapes are defined for arbitrary Hausdorff compacta. Maps between such systems are defined as well as a notion of homotopy of such maps. This homotopy relation classifies maps between ANR-systems and these classes are called shape maps. In both cases the description of shape involves some external elements like ambient spaces where to embed the compacta or ANR-expansions.The aim of this paper is to present a new description of the shape category of compacta in which no use is made of such external elements. Our description is intrinsic in the sense that we only use maps (or sequences of maps) between compacta as opposed to the usual descriptions where maps are defined between some objects associated to the compacta. Our maps are multivalued. We introduce the notion of a multi-net between compacta which consists of a sequence of upper-semicontinuous multivalued maps progressively smaller, becoming more and more similar to a single-valued map. This notion is related to

-connectedness, finite approximations, shape theory and coarse graining in hyperspaces

“…Then the sequence of maps (q A n ) : X −→ 2 X U represents the identity shape morphism on X . The representation quoted in the previous proposition is in the sense described first by Sanjurjo in [47] and reinterpreted later in [9]. Other results with influence in our future work -and intrinsically related to our Example 1 in the introduction -are those due to Latschev in [48].…”

Global topological properties of the Hopf bifurcation

“…We use, in particular, some wellknown facts related to properties of stability and attraction of flows. For information about the topological and the dynamical aspects of these theories we refer the reader to the papers and books [2,4,6,7,14,24,27,28,32,34,35,40] (see also [3,[11][12][13]20,30,31,[36][37][38] for applications of the theory of shape to dynamical systems). The paper [44] by Viana explores the relations between the study of attractors and that of main bifurcation mechanisms.…”

The Hausdorff Metric and Classifications of Compacta

“…Sanjurjo asked us about the possibility of having a better description of shape following his line. In particular, he enquired about the possibility of changing upper semicontinuity to continuity (upper and lower [18]) in the description given in [21]. We answer the question positively, for the realm of locally connected compacta.…”

“…The first part is motivated by J. M. R. Sanjurjo [21] and the second part by T. A. Chapman [9]. In [21] the author describes shape theory in terms of sequences of upper semicontinuous multivalued maps with decreasing diameters on its images.…”

“…In [21] the author describes shape theory in terms of sequences of upper semicontinuous multivalued maps with decreasing diameters on its images. Sanjurjo asked us about the possibility of having a better description of shape following his line.…”